Concept

Transient

The early part of an orbit, before it has settled onto whatever it settles onto. Discarding it is what makes a bifurcation diagram legible, and how long it lasts is a separate question from where it ends up.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

the logistic map at 3.2, iterated from 0.2. A map drawn as a curve with the diagonal across it, and the staircase that iterating it produces.

The staircase that shows the whole orbit

Take a number, feed it to a rule, feed the answer back in. There is a way of drawing that on the rule's own graph which turns the entire future of a starting point into a shape — and the shape is legible.

dynamics · Iteration

The logistic map's bifurcation diagram, 2.4 to 4. For each parameter, the values the orbit settles into, plotted as a column of points.

The road paved with doublings

Turn one dial slowly and watch what a map settles into. It settles on a point, then on two points, then four, then eight — faster and faster, and the doublings run out at a parameter that is finite.

dynamics · Period-doubling

Intermittency at r = 1 + √8 − 0.0003. A time series of 600 steps of the logistic map just below the period-three window. Long stretches that look like a cycle of three, shaded, alternate with irregular bursts; there are 6 such stretches here.

The window that opens with a stutter

The period-three window does not fade in. At r = 1 + √8 a cycle of three appears out of nothing, and just before it does, the chaotic orbit keeps imitating the cycle that is not there yet — for twenty steps, then fifty, then hundreds, in quiet stretches whose length grows as one over the square root of the distance to the window.

dynamics · Period-doubling

Named alongside it

The objects these essays reach for when they reach for this one.

Logistic map Periodic orbit Attractor Bifurcation Chaos Fixed point Self-similarity Cobweb Intermittency Iteration Orbit Period-doubling

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